Antiproximinal Sets in Banach Spaces of Continuous Vector-Valued Functions1

A closed nonvoid subset Z of a Banach space X is called antiproximinal if no point outside Z has a nearest point in Z. The aim of the present paper is to prove that, for a compact Hausdorff space T and a real Banach space E, the Banach space C T E , of all continuous functions defined on T and with values in E, contains an antiproximinal bounded closed convex body. This extends a result proved by V. S. Balaganskii (1996, Mat. Zametki 60, 643–657) in the case E = .